This paper describes several mechanical balancers, based on the planetary gear trains known as the hypotrochoid and epitrochoid trains, for reducing or eliminating second-order out-of-balance in mechanical systems. It is shown that by proper arrangement of the planetary gear trains, a balancer can be obtained for the elimination of second-order shaking forces or second-order shaking moments or a combination of both shaking forces and moments. The advantage of this type of balancer is that the carrier of the gear train needs only to run at the primary speed of the mechanical system to be balanced. Therefore, the balancer can be designed to be concentric with the primary rotating shaft of the machine using the primary shaft as the carrier. For example, for the balance of the second-order shaking force of an in-line four-cylinder four-stroke internal combustion piston engine, the balancer can be placed on the third main bearing or one on each of the second and fourth main bearings.